PowerPoint Presentation

CS 590Term Project

Epidemic model on FacebookChoungRyeol LEE, Shubham Agrawal, Ashwin JiwaneFacebook (partial) Network

Source: Facebook ego network, Stanford Network Analysis ProjectData Limitation and ProcessingIt is infeasible for us to access (and handle) the complete Facebook dataAnalysis is done on partial dataset obtained from Stanford Network Analysis ProjectThe original data is the directed ego-network (without ego) of 10 nodes which we had to reconstruct, i.e. make it undirected and add the ego-edges

Ego-network: 3What is Ego Network?

Source: Slides by Giorgos Cheliotis, National University of SingaporeNetwork CharacteristicsNetwork CharacteristicValueNumber of nodes (n)3963Number of edges (m)88156Number of cluster (c)1Minimum degree (dmin)2Maximum degree (dmax)1034Average degree (d)22.245Average path length (l)3.776Diameter (D)8Global clustering coefficient (cc)0.5212Maximum clique size57Centrality MeasuresWeightCentrality MeasuresNodeWeightedEigenvector2160Pagerank1641Closeness100Betweenness100Degree100Non-WeightedEigenvector1868Pagerank3381Closeness100Betweenness100Degree100Basic AnalysisCentrality MeasureFacebook InterpretationPagerankIt is very likely to visit his profile in random surfing starting from anyone elses profileEigenvectorThis person has influential (or social) friendsBetweennessThis person is an important connection between different peopleClosenessThis person uses minimum amount of mutual friends link to connect to anyone elseDegreeThis person has maximum number of friendsObservations:The graph follows the Small World Phenomenon as the average path length is 3.776 but it is not a Scale-Free network since it doesnt follow Power-Law7Power-LawFriendship StrengthIn FB, possible ways to measures friendship:Mutual friendsCommon biography (location, education, etc)Mutual interests (pages, likes, etc)Common social groupsDue to limitation of data, we considered only Mutual Friends as the weighing measureCosine SimilarityCosine similarity measures the normalized number of common friendsBasic principle is to take the cosine of the vectors (rows) from adjacency matrixIn study network:Maximum cosine value = 0.961454 Minimum cosine value = 0.003408 Epidemic ModelsSI and SIR Model:A node is susceptible to infected node with certain probabilityYou repost/share from friendsSI Model:Once a node is infected, it remains infectedPost remains active on the wall SIR Model:Once a node is infected, it remains infected for certain time period Post gets inactive after certain time period

Model SimulationSimulated epidemic model on the graphPre-infected a particular nodeCompared the results with different nodes of importanceChecked for the time steps required for complete cascade in SI modelChecked for the time steps required to reach stable condition in SIR modelStable condition means no more node is getting infected due to died nodes12Model SimulationModel assumptions:Probability of infectionDiscrete time intervalsAssumed two scenarios of probability:Function of weightSimilar to Top News postsIndependent of weightSimilar to Most Recent postsSIR Model ResultsFunctionWWW0.20.20.2ModelSISISISISISIImportanceEigenPageDegreeEigenPageDegreeNode2160164110018683381100TimeStepFreqFreqFreqFreqFreqFreq1111111216717221916812023332544766934912766794269133403463225669541338789872464925646598723482771507743535242766376630986297022494129062279549631164962571141063316910978381611273935511715121221311030413113123557150214116215111578631162811178182190201SIR Model ResultsSIR InterpretationUnweighted graph (p=0.2):Degree: steepest curve, infects less people, EigenVector: steep curve, infects most peoplePagerank: grows slowest, infects more peopleWeighted graph(p=0.2):Degree: steepest curve, infects less peopleEigenVector: grows slowest, infects most peoplePagerank: grows slow, infects more people, better than eigenvector due to weights

SI Model Results2160 Weig, 1641 Wpage, 100 Wdegree; 1868 Eig, 3381 Page, 100 - Degree17Currently working on..Quarantine Strategy:Choose the nodes to quarantine at a certain time interval such that they dont affect othersAccount blocked (reported as spam)Vaccination Strategy:Choose the nodes to vaccinate, i.e. make them safe from certain viral, such that epidemic doesnt flow through themSpam filterObjective is to minimize the cost of prevention and/or precaution with the aim of curing epidemicQuarantine We are picking up people based on the time they have been infected and their importance in the networkVaccination We are saving people based on the number of infected neighbors and their importance in the network18Communities in Facebook networkHeld together by some common interests and ideas of a large group of people in FacebookAny one person may be part of many communities which are overlapping and nested structureGroups within social networks might highly correspond to social units or communities in realityA subset of Facebook users within the graph such that connections between the users are denser than connections with the rest of the network.One person has only one community

19Reviews of Community DetectionTwo methods for discovering groups in networksGraph partitioningPre-fixed number of parts by minimizing cut edge Computation load(NP-Hard)Community structure detectionSuitable for the structure of large-scale network dataProvides information on topology of the network

Two approaches really want to address the same question with somewhat different means. 20Community Detection using iGraphAlgorithms in IgraphOptimal communitiesBasic frameworkInfomap92 communitiesCompressing the description of information flows on networks.Leading Eigenvector18 communitiesCalculation of Leading non-negative eigenvector of the modularity matrix of the graph and distributions of vertices by the sign of eigenvectorLabel Propagation57 communitiesLabeling with unique labels and updating by majority voting in the neighbors of the vertexMultilevel17 communitiesContribution to modularity with sequential changes of assignment of nodesThank You